The ellipsoidal wave equation is the name given to the ordinary differential equation which arises when the wave equation (Helmholtz equation) is separated in ellipsoidal co-ordinates. In this paper, solutions of the equation are expressed as Neumann series (series of Bessel functions of increasing order).

A matrix analysis is derived, using the concept of finite elements, for beam-columns of continuously varying cross-section and “small” initial curvature subject to tangential and normal forces acting at discrete points along the centroidal axis.

The relationships between loads applied to the ends of the member and their corresponding deformations are established in the form of stiffness matrices, which are the basis of the Equilibrium Method of Analysis of linear or non-linear elastic plane frames composed of such membeis. In addition, the end loads induced by the tangential and normal forces for various types of boundary conditions are obtained.

Finally, it is shown that the buckling load may be calculated by the determination of the lowest eigenvalue of a certain matrix, provided there is no lateial translation of the ends of the member.

In the case of Boolean matrices a given eigenvector may have a variety of eigenvalues. These eigenvalues form a sublattice of the basic Boolean algebra and the structure of this sublattice is investigated. Likewise a given eigenvalue has a variety of eigenvectors which form a module of the Boolean vector space. The structure of this module is examined. It is also shown that if a vector has a unique eigenvalue λ, then λ satisfies the characteristic equation of the matrix.

The traditional method of solution to problems in linear viscoelasticity theory involves the direct application of the Laplace transform to the relevant field equations and boundary conditions. If the shape of the body under consideration or the type of boundary condition specified at a point or both vary with time then this method no longer works. In this paper we investigate the applicability of stress function solutions to this situation. It is shown that for time-dependent ablating regions a generalization of the Papkovich Neuber stress function solution of elasticity holds. As an example the stress and displacement fields are calculated for the problem of an infinite viscoelastic body with a spherical ablating stress free cavity and prescribed time-dependent stresses at infinity.

An investigation has been made of the condensation nuclei created by an electric field in a N2 – H2O mixture. These nuclei are distinguished by the fact that they induce condensation in a vapour which is only 4 per cent supersaturated.

An explanation of these phenomena is found in the presence of nitrogen dioxide vapour, one of the products of reactions induced by the field, exerting a small pressure PNO2<1O−6 mm. Hg.

The observations are consistent with the assumption that the nuclei are created in the reaction, 2NO2 + H2O ⇌ HNO2 + HNO3. It is believed that the reason for the requirement of a more than critically supersaturated vapour is that this must be the condition for the nuclei forming reaction to proceed.

Once the nuclei have been created, any additional quantity of NO2 collected from the vapour forms acid molecules which promote condensation from a vapour which is not necessarily supersaturated.

Drops formed by these nuclei contain a significant quantity of HNO2 + HNO3, so that, unlike drops of pure water, they are stable against reevaporation in a vapour, the relative humidity of which is <100 percent.

Using a technique due to Macbeath (Jack and Macbeath 1959) this paper gives what the author hopes is a shorter and easier presentation of the evaluation of certain Jacobians of matrix transformations which have occurred in statistics and the theory of quadratic forms.

The types of modes which may exist in an infinite parallel-plate waveguide with a centrally-placed unidirectionally conducting screen are studied. The effect of bifurcating an infinite parallel-plate waveguide by such a screen in the region x>O is investigated when a transverse electric mode is incident on the bifurcated region. The problem is solved by the Wiener-Hopf method, and expressions are derived for the amplitudes and phases of the reflected and transmitted modes. It is found that the transmitted field contains a hybrid wave and a slow wave, and the reflected field contains transverse electric and transverse magnetic waves.

A graph consists of a set of vertices some pairs of which are joined by a single edge. A tree is a graph with the property that each pair of vertices is connected by precisely one path, i.e., a sequence of distinct vertices joined consecutively by edges. The complexity c of a graph G(n, k) with n vertices and k edges is the number of trees with n vertices which are subgraphs of G(n, k). The distribution of c over the class of all graphs G(n, k) is of physical interest because it throws light on the classical many-body problem. (See, e.g. [9].) Ford and Uhlenbeck [3] gave numerical data which suggested that the distribution of c tends to normality for increasing n if k is near No moments higher than the first were known in general and they remarked in [4] that even “the second would be worth knowing”. The main object in this paper is to derive a formula for the second moment of c.

Summary. This paper is concerned with an infinite plate of homogeneous isotropic elastic material in a state of generalised plane stress and having a circular hole with boundary γ divided into two parts. Over one part of γ the stresses are zero; over the other the shear stress is zero and the normal displacement is specified. The problem corresponds to a smooth loose rigid pin pressed against the edge of a circular hole in an infinite plate.